An Optimal Control Theory for the Traveling Salesman Problem and Its Variants

We show that the traveling salesman problem (TSP) and its many variants may be modeled as functional optimization problems over a graph. In this formulation, all vertices and arcs of the graph are functionals; i.e., a mapping from a space of measurable functions to the field of real numbers. Many variants of the TSP, such as those with neighborhoods, with forbidden neighborhoods, with time-windows and with profits, can all be framed under this construct. In sharp contrast to their discrete-optimization counterparts, the modeling constructs presented in this paper represent a fundamentally new domain of analysis and computation for TSPs and their variants. Beyond its apparent mathematical unification of a class of problems in graph theory, the main advantage of the new approach is that it facilitates the modeling of certain application-specific problems in their home space of measurable functions. Consequently, certain elements of economic system theory such as dynamical models and continuous-time cost/profit functionals can be directly incorporated in the new optimization problem formulation. Furthermore, subtour elimination constraints, prevalent in discrete optimization formulations, are naturally enforced through continuity requirements. The price for the new modeling framework is nonsmooth functionals. Although a number of theoretical issues remain open in the proposed mathematical framework, we demonstrate the computational viability of the new modeling constructs over a sample set of problems to illustrate the rapid production of end-to-end TSP solutions to extensively-constrained practical problems.Comment: 24 pages, 8 figure

Polarimetric modeling of corotating interaction regions (CIRs) threading massive-star winds

Massive star winds are complex radiation-hydrodynamic (sometimes magnetohydrodynamic) outflows that are propelled by their enormously strong luminosities. The winds are often found to be structured and variable, but can also display periodic or quasi-periodic behavior in a variety of wind diagnostics. The regular variations observed in putatively single stars, especially in UV wind lines, have often been attributed to corotating interaction regions (CIRs) like those seen in the solar wind. We present light curves for variable polarization from winds with CIR structures. We develop a model for a time-independent CIR based on a kinematical description. Assuming optically thin electron scattering, we explore the range of polarimetric light curves that result as the curvature, latitude, and number of CIRs are varied. We find that a diverse array of variable polarizations result from an exploration of cases. The net polarization from an unresolved source is weighted more toward the inner radii of the wind. Given that most massive stars have relatively fast winds compared to their rotation speeds, CIRs tend to be conical at inner radii, transitioning to a spiral shape at a few to several stellar radii in the wind. Winds with a single CIR structure lead to easily identifiable polarization signatures. By contrast allowing for multiple CIRs, all emerging from a range of azimuth and latitude positions at the star, can yield complex polarimetric behavior. Although our model is based on some simplifying assumptions, it produces qualitative behavior that we expect to be robust, and this has allowed us to explore a wide range of CIR configurations that will prove useful for interpreting polarimetric data.Comment: accepted to A&

Migration-selection balance at multiple loci and selection on dominance and recombination

A steady influx of a single deleterious multilocus genotype will impose genetic load on the resident population and leave multiple descendants carrying various numbers of the foreign alleles. Provided that the foreign types are rare at equilibrium, and that all immigrant genes will eventually be eliminated by selection, the population structure can be inferred explicitly from the deterministic branching process taking place within a single immigrant lineage. Unless the migration and recombination rates were high, this simple method was a very close approximation to the simulated migration-selection balance with all possible multilocus genotypes considered.Comment: includes 6 figures and a Supporting Information. Mathematica notebook where the numerical results were obtained is available upon reques

Fast Mesh Refinement in Pseudospectral Optimal Control

Mesh refinement in pseudospectral (PS) optimal control is embarrassingly easy --- simply increase the order $N$ of the Lagrange interpolating polynomial and the mathematics of convergence automates the distribution of the grid points. Unfortunately, as $N$ increases, the condition number of the resulting linear algebra increases as $N^2$; hence, spectral efficiency and accuracy are lost in practice. In this paper, we advance Birkhoff interpolation concepts over an arbitrary grid to generate well-conditioned PS optimal control discretizations. We show that the condition number increases only as $\sqrt{N}$ in general, but is independent of $N$ for the special case of one of the boundary points being fixed. Hence, spectral accuracy and efficiency are maintained as $N$ increases. The effectiveness of the resulting fast mesh refinement strategy is demonstrated by using \underline{polynomials of over a thousandth order} to solve a low-thrust, long-duration orbit transfer problem.Comment: 27 pages, 12 figures, JGCD April 201

The role of plant functional trade-offs for biodiversity changes and biome shifts under scenarios of global climatic change

The global geographic distribution of biodiversity and biomes is determined by species-specific physiological tolerances to climatic constraints. Current vegetation models employ empirical bioclimatic relationships to predict present-day vegetation patterns and to forecast biodiversity changes and biome shifts under climatic change. In this paper, we consider trade-offs in plant functioning and their responses under climatic changes to forecast and explain changes in plant functional richness and shifts in biome geographic distributions. <br><br> The Jena Diversity model (JeDi) simulates plant survival according to essential plant functional trade-offs, including ecophysiological processes such as water uptake, photosynthesis, allocation, reproduction and phenology. We use JeDi to quantify changes in plant functional richness and biome shifts between present-day and a range of possible future climates from two SRES emission scenarios (A2 and B1) and seven global climate models using metrics of plant functional richness and functional identity. <br><br> Our results show (i) a significant loss of plant functional richness in the tropics, (ii) an increase in plant functional richness at mid and high latitudes, and (iii) a pole-ward shift of biomes. While these results are consistent with the findings of empirical approaches, we are able to explain them in terms of the plant functional trade-offs involved in the allocation, metabolic and reproduction strategies of plants. We conclude that general aspects of plant physiological tolerances can be derived from functional trade-offs, which may provide a useful process- and trait-based alternative to bioclimatic relationships. Such a mechanistic approach may be particularly relevant when addressing vegetation responses to climatic changes that encounter novel combinations of climate parameters that do not exist under contemporary climate

Separation and identification of dominant mechanisms in double photoionization

Double photoionization by a single photon is often discussed in terms of two contributing mechanisms, {\it knock-out} (two-step-one) and {\it shake-off} with the latter being a pure quantum effect. It is shown that a quasi-classical description of knock-out and a simple quantum calculation of shake-off provides a clear separation of the mechanisms and facilitates their calculation considerably. The relevance of each mechanism at different photon energies is quantified for helium. Photoionization ratios, integral and singly differential cross sections obtained by us are in excellent agreement with benchmark experimental data and recent theoretical results.Comment: 4 pages, 5 figure

Providing Feedback Following Leadership Walkrounds is Associated with Better Patient Safety Culture, Higher Employee Engagement and Lower Burnout

Background There is a poorly understood relationship between Leadership WalkRounds (WR) and domains such as safety culture, employee engagement, burnout and work-life balance. Methods This cross-sectional survey study evaluated associations between receiving feedback about actions taken as a result of WR and healthcare worker assessments of patient safety culture, employee engagement, burnout and work-life balance, across 829 work settings. Results 16 797 of 23 853 administered surveys were returned (70.4%). 5497 (32.7% of total) reported that they had participated in WR, and 4074 (24.3%) reported that they participated in WR with feedback. Work settings reporting more WR with feedback had substantially higher safety culture domain scores (first vs fourth quartile Cohen’s d range: 0.34–0.84; % increase range: 15–27) and significantly higher engagement scores for four of its six domains (first vs fourth quartile Cohen’s d range: 0.02–0.76; % increase range: 0.48–0.70). Conclusion This WR study of patient safety and organisational outcomes tested relationships with a comprehensive set of safety culture and engagement metrics in the largest sample of hospitals and respondents to date. Beyond measuring simply whether WRs occur, we examine WR with feedback, as WR being done well. We suggest that when WRs are conducted, acted on, and the results are fed back to those involved, the work setting is a better place to deliver and receive care as assessed across a broad range of metrics, including teamwork, safety, leadership, growth opportunities, participation in decision-making and the emotional exhaustion component of burnout. Whether WR with feedback is a manifestation of better norms, or a cause of these norms, is unknown, but the link is demonstrably potent

Multiphoton detachment of electrons from negative ions

A simple analytical solution for the problem of multiphoton detachment from negative ions by a linearly polarized laser field is found. It is valid in the wide range of intensities and frequencies of the field, from the perturbation theory to the tunneling regime, and is applicable to the excess-photon as well as near-threshold detachment. Practically, the formulae are valid when the number of photons is greater than two. They produce the total detachment rates, relative intensities of the excess-photon peaks, and photoelectron angular distributions for the hydrogen and halogen negative ions, in agreement with those obtained in other, more numerically involved calculations in both perturbative and non-perturbative regimes. Our approach explains the extreme sensitivity of the multiphoton detachment probability to the asymptotic behaviour of the bound-state wave function. Rapid oscillations in the angular dependence of the $n$-photon detachment probability are shown to arise due to interference of the two classical trajectories which lead to the same final state after the electron emerges at the opposite sides of the atom when the field is close to maximal.Comment: 27 pages, Latex, and PostScript figures fig1.ps, fig2.ps, fig3.ps, accepted for publication in Phys. Rev.